Slip optimization on arbitrary 3D microswimmers: a reduced-dimension and boundary-integral framework

TL;DR

This paper introduces a computational framework that optimizes slip velocities for arbitrary 3D microswimmers in viscous fluids, reducing complex PDE problems to low-dimensional, cost-effective calculations.

Contribution

It develops a boundary-integral method leveraging linearity and reciprocity to efficiently optimize microswimmer slip profiles across arbitrary geometries.

Findings

Validated the method's accuracy with various shapes.

Generated optimal slip profiles and trajectories.

Analyzed effects of geometrical symmetries on motion.

Abstract

This article presents a computational framework for determining the optimal slip velocity of a microswimmer with arbitrary three-dimensional geometry suspended in a viscous fluid. The objective is to minimize the hydrodynamic power dissipation required to maintain unit speed along the net swimming direction. By exploiting the linearity of the Stokes equations and the Lorentz reciprocal theorem, we derive an explicit linear operator that maps the tangential surface slip velocity to the resulting rigid-body translational and rotational velocities, effectively decoupling the hydrodynamic boundary value problem from the optimization loop. The a priori infinite-dimensional search space for the slip optimization is reduced to the finite dimension $r$ of rigid-body motions by finding an appropriate subspace of the operator's domain. This reduces the PDE-constrained optimization to a low-dimensional programming problem that can be solved at negligible computational cost once the system matrices are assembled. The optimization algorithm requires 2$r$ auxiliary flow problems that are solved numerically using a high-order boundary integral method. We validate the accuracy of the proposed method and present optimal slip profiles and swimming trajectories for a variety of microswimmer shapes. We investigate the effect of some common geometrical symmetries of the swimmer shape on the resulting optimal motion, and in particular present a modified version of the slip optimization algorithm for axisymmetric shapes, where tangential rigid-body velocities may occur